A strong personality.
What kind of person is Hitsugaya anyway? Well, at first glance, he seems to be an acerbic condescending little snot with a perpetual glare forever etched into his tiny face. He has all the regal attitude of a long-suffering teenager forced to deal with the authority of adults whether he likes it or not - and he definitely does not. He is sixty-two pounds of pure snark. This is the first glimpse we see of him.
❝Here we go again, stupid old men bickering.❞
This comes off as an arrogant remark, while it really is just indicative of how easily annoyed Hitsugaya is. He takes his role as a Captain very seriously and expects the same of others. He doesn't have the patience for those who goof off or unnecessarily waste time. An argument over who should take blame is less productive than one about what actions to take to solve a problem. He would rather figure out a solution and implement it quickly instead of waiting around for others to work out their petty issues.
To that end, it is worth noting that Hitsugaya is highly intelligent. He is more than capable of learning and drawing his own conclusions without the explanation of others, though he too is human(-spirited?) and capable of making mistakes. His intelligence is displayed and referenced many times throughout the series, as he is known to have graduated the shinigami academy within one year, a record shared with only one other person. He is one of very few who became suspicious of the circumstances regarding Kuchiki Rukia's trial and sentencing, and set out on an investigation of his own to discover the truth of what was happening. He was highly suspicious of Gin Ichimaru, as he should have been. However, his perceptiveness can be used against him, as he was so intently focused on Gin that he didn't realize who the real puppet master was, playing into his trap all along.
This intelligence also allows him to think quite flexibly and adapt to all manner of situations. At one point, when his bankai is stolen from him, he develops new and unorthodox methods of using his weaker shikai power for greater effect. In combat, Hitsugaya is collected and calculating, constantly forming plans and attacks no matter the foe, and it takes extreme measures to distract him from his opponent. He is always assessing the situation, always has an appropriate reaction to a move even if he is in pain or faced with overwhelming power. As long as he isn't down, there must be something he can do to turn the tables. It is this flexible and informed decision-making that allows him to defeat his enemies.
One of Hitsugaya's good traits is the consideration he has for those around him. He isn't a pushy or nosy person, staying well out of other people's matters, but when someone important to him is suffering, he will do what he can to help from the background. When his lieutenant, Matsumoto Rangiku, is emotionally shaken by the attitude and actions of her childhood friend, Hitsugaya allows her to rest far longer than he normally would have and then treating her entirely normally when he awake. The only reason she knows it is intentional is because he almost completed her entire share of paperwork. It is his awkward way of showing his consideration. He also waits silently after Kira goes to makes amend with Matsumoto, quietly watching over them to make sure they are getting along before dismissing himself. He doesn't say anything when he knows that words won't help, but he silently supports those important to him and keeps an eye on them when they could be vulnerable. It is the kind of quiet consideration of a person who doesn't seek thanks or recognition. He would prefer it not be mentioned at all, because he is easily flustered when it comes to genuine emotion.
Which brings me to his social and emotional awkwardness. Based on his childhood, where he was ousted by nearly the entire neighborhood, Hitsugaya has little experience with social interactions. His only relationships were limited to his grandmother and Hinamori, and his friendship with her is one of an inordinate amount of teasing. He is incredibly socially inept, and the only way he knows how to communicate with others is with sharp words and glares. He doesn't quite know how to handle the genuine emotions of others directed at him, or his own when they are. He seemed astounded when Hinamori and Matsumoto organized a small firework show for his birthday, and finds thanking them to be a difficult task.
❝Thank you... Matsumoto.❞
Hitugaya is straightforward, certainly, but he doesn't quite know how to handle when other people are straightforward with him. He finds affection sent his way difficult to comprehend, likely because of a childhood full of rejection. He has hardened his heart, and is only prepared for the bad things. He isn't at all prepared for the moments when he is loved, or cared for, or treasured. He simply hides his face and his emotions and mumbles a response before starting with the teasing again. And it is adorable. He knows what needs to be said, he just has too little experience with actually saying it. Certainly, it must be frustrating to have an emotionally stunted Captain who is so guarded that he forgets to ask for help, but Matsumoto is more than up to the task of dealing with him, and she is not the type of person who would do so if he wasn't important to her. And the more time he spends with others who are important to him that cherish him in turn, the less awkward he becomes. Someone caring about him becomes less surprising, and thanking them or relying on them becomes easier. And allowing himself to openly care for someone else becomes easier too.
However he changes, he will still always be known for his general grumpiness. He has relatively short fuse, a rather fiery trait for someone with an ice affinity, and has little patience. Poor Hitsugaya is surrounded by easygoing, somewhat simple-minded subordinates who rarely listen to what he has to say despite their difference in rank. When he has to lead a group of shinigami to Ichigo's hometown, he is very quickly overwhelmed by all of their strong, ridiculous personalities. They don't disrespect him by any means, as they will listen to every word he says when the situation is serious. But if its not, then he's most likely out of luck. Their shenanigans very quickly exhaust him and he usually ends up frustrated and trying desperately to ignore them, if he's not yelling at them repeatedly.
One of the things that gets him worked up is being treated like a child. I mean, he is one, technically, but he doesn't want to be treated as one. He worked very hard for his position, and so whenever someone fails to call him Captain, he is quick to anger and correct them. As if in defiance to his treatment, he doesn't refer to any of the other Captains by their titles. They seem to care less about it than he does. While he is angered when he is treated as a child, there are a few people he will tolerate doing so, to varying levels. His opponents often treat him as such, and he seemingly brushes it off as if he knows it means they are underestimating him. He seems doomed to be eternally frustrated with the fact that he is treated like a child when the ones around him act more juvenile than he does.
One of Hitsugaya's strongest character traits is his passion. It is another of his oddly fiery traits that contrasts starkly with his element. He is fiercely loyal to those he cares for, and so when Hinamori becomes a victim of a villain's heinous plot, he quickly becomes obsessed with helping her.
❝I'm going to save Hinamori.❞
Even when she turns her blade on him, Hitsugaya's passion to save her never falters. Whatever actions she takes against him at that point are not her own, they are the actions of the enemy who forced her hand. His devotion to her is intense and unbreakable. It might take someone a long time to find a place in his heart, but when they do he will never let them go. He is completely overcome with the need to protect those he cares for from anyone who means them harm, by any means necessary.
However, passion is a double-edged sword, as it can lead to other strong emotions, some a little more problematic than others. One of Hitsugaya's greatest internal obstacles is his own rage. His anger makes him predictable and can be used against him, to manipulate him, as we see in his attempts to fight Aizen. This villain knows exactly how to make Hitsugaya relinquish control to his rage, then use that precise moment to stop him in his tracks. This happens more than once, so it is clearly a recurring issue that needs work, but unfiltered rage is incredibly difficult to get under control.
It is important to note that Hitsugaya never becomes infuriated when something is said about or done to him. It is only when those important to him are threatened or harmed that he truly sees red. This is why his passion is one of his strongest traits, whether it is beneficial or not. It is huge part of him, both a strength and a weakness. One of the most difficult parts of being human is that once you welcome someone into your heart, they can then be used to help you or as a weapon to hurt you. Even knowing that can't change the effect it has on you, or prepare you for the moments to come. This is true even for fictional sword-toting spirits that save the world on multiple occasions.
Hitsugaya's greatest character trait, in my opinion, is his complete determination. He has an intense dedication toward becoming stronger by any means, and the determination to not be held back by extenuating circumstances. He doesn't wallow in self-pity at the loss of his bankai, though he is occasionally plagued by the stray dark thought here and there - he dedicates himself to finding new ways to become stronger, to stay in the fight. Doggedly pursuing the truth behind Rukia's situation is another example of his determination and dedication. He is not the type of person to give up easily, no matter the circumstance.
Even if his actions have no effect, or his ideas get him nowhere, or his plan fails, Hitsugaya still persists. He never once gives up on any of his goals. He doesn't give in when the odds are unfairly against him. He does not give in to despair, to exhaustion, or to pain, often pushing himself to his limit. He is always trying
. He never stops trying, and never lets his failure affect his determination to do so. No matter what happens, no matter who he has to fight, no matter how he is perceived, he never stops trying. It is a quality I find incredibly admirable, because anyone knows just how easy it is to simply stop trying if the going becomes too rough. But Hitsugaya persists, and is determined to succeed even if the entire world is against him.
Despite his determination and passion, Hitsugaya truly treasures peace above everything else. Though he is constantly annoyed by the antics of his subordinates and is always trying to get Matsumoto to do her share of the paperwork, he would prefer all of that over real combat or war any day. Getting angry because his lieutenant slipped away to get drunk is far more appealing than watching her bleed out on the ground beside him. Getting frustrated over trivial matters is a luxury, and he recognizes that. While he will never stop trying to become stronger, never losing his desire to protect those important to him, he would rather have nothing to protect them from. When he speaks about the peace they have achieved at the end of the series, it is one of the only times we ever see him smile, stating that it is good that they have no cause to use their powers. He wants the power to protect, but would be far more satisfied in a world where he doesn't need to protect anyone at all.